For an open subset $U$ of the algebraic variety $X$, a moment map is a morphism $\varphi:U\rightarrow T$ to a split multiplicative torus $T:={\mathbb{G}}_{m}^{r}$ over $K$. The tropicalization of $\varphi$ is

Obviously, this is a continuous map with respect to the topology on the analytification ${U^{\rm an}}$. Note that our moment maps are algebraic which differs from the moment maps in [CD12] which are defined analytically.

We say that the moment map $\varphi^{\prime}:U^{\prime}\rightarrow T^{\prime}$ of the open subset $U^{\prime}$ of $X$ refines the moment map $\varphi:U\rightarrow T$ if $U^{\prime}\subset U$ and if there is an affine homomorphism $\psi:T^{\prime}\rightarrow T$ of the multiplicative tori such that $\varphi=\psi\circ\varphi^{\prime}$ on $U^{\prime}$. Here, an affine homomorphism means a group homomorphism composed with a (multiplicative) translation on $T$. This group homomorphism induces a homomorphism $M\rightarrow M^{\prime}$ of character lattices. Its dual is the linear part of an integral affine map ${\rm Trop}(\psi):N^{\prime}_{\mathbb{R}}\rightarrow N_{\mathbb{R}}$ such that ${\varphi_{\rm trop}}={\rm Trop}(\psi)\circ\varphi_{\rm trop}^{\prime}$ on $(U^{\prime})^{\rm an}$.

If $\varphi_{i}:U_{i}\rightarrow T_{i}$ are finitely many moment maps of non-empty open subsets $U_{i}$ of $X$ with $i\in I$, then $U:=\cap_{i}U_{i}$ is a non-empty open subset of $X$ and $\varphi:U\rightarrow\prod_{i}T_{i},x\mapsto(\varphi_{i}(x))_{i\in I}$ is a moment map which refines every $\varphi_{i}$. Moreover, it follows easily from the universal property of the product that every moment map $\varphi^{\prime}:U^{\prime}\rightarrow T^{\prime}$ which refines every $\varphi_{i}$ refines also $\varphi$.